My impression is that the name matching was done on the airline side. The rental company doesn't have access to the airline's database, so they wouldn't know what the name on the FF account is. But when they submit their data to United/AA/DL/etc., they also include the name on the contract, and then the airline rejects it if the name doesn't match.

The question I've never figured out the solid answer to, though, is how strict they are with name-matching. In other contexts, I've seen the name matching be quite loose, to the point that if even only the last name matches, it works fine. If that's the case with the OP, who knows--it might work. But I dont know any of the particulars between Avis and United.